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Algebrability of the set of everywhere surjective functions on \(\mathbb C\). (English) Zbl 1130.46013

Summary: We show that the set \(\mathcal L\) of complex-valued everywhere surjective functions on \(\mathbb C\) is algebrable. Specifically, \(\mathcal L\) contains an infinitely generated algebra every nonzero element of which is everywhere surjective. We also give a technique to construct, for every \(n\in\mathbb N\), \(n\) algebraically independent everywhere surjective functions, \(f_1,f_2,\dots,f_n\), so that for every non-constant polynomial \(P\in\mathbb C[z_1,\dots,z_n]\), \(P(f_1,f_2,\dots,f_n)\) is also everywhere surjective.

MSC:

46E25 Rings and algebras of continuous, differentiable or analytic functions
15A03 Vector spaces, linear dependence, rank, lineability
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Full Text: Euclid